Saturday, November 15, 2014

Kirk Durston Does Mathematics!

Kirk Durston is a local evangelical Christian who likes to construct unconvincing arguments for his faith. Every few years he trots them out at my university.

Here is one of his more recent attempts, a discussion of infinity. Not surprisingly, it is a confused mess.

Durston's argument is based, in part, on a distinction that does not really exist: between "potential infinity" and "completed infinity" or "actual infinity". This is a distinction that some philosophers love to talk about, but mathematicians generally do not.* You can open any contemporary mathematical textbook about set theory, for example, and not find these terms mentioned anywhere. Why is this? It's because mathematicians understand the subject well, but -- as usual -- many philosophers are extremely muddled thinkers when it comes to infinity.

Here is how Durston defines "potential infinity": "a procedure that gets closer and closer to, but never quite reaches, an infinite end". So, according to Durston, a "potential infinity" is not a set but a "procedure". Yet the very first example that Durston gives is "the sequence of numbers 1,2,3, ... gets higher and higher but it has no end". The problem should be clear: a "sequence" is not a "procedure"; a (one-sided infinite) sequence over a set S is a mapping from the non-negative integers to S. From the beginning, Durston is quite confused. His next example is "the limit of a function as x approaches infinity". But a "limit" is not a "procedure", either. Durston also doesn't seem to understand that limits involving the symbol ∞ can be restated to avoid it entirely; the ∞ in a limit is a shorthand that has little to do with infinite sets at all.

He defines "actual infinity" or "completed infinity" as "an infinity that one actually reaches", which doesn't seem to have any actual meaning that I can divine. But then he says that "actual infinity" or "completed infinity" is "just one object, a set". Fair enough. Now we know that for Durston, an "actual" or "completed" infinity is a set.
But what does it mean for a set to "reach" something? And if we consider the set of natural numbers, for example, what does it mean to say that it "reaches" infinity? After all, the set of natural numbers N contains no number called "infinity", so if anything, we should say that N does not "reach" infinity.

But then he goes on to say "First, a completed countable infinity must be treated as a single ‘object’." This is evidently wrong. For Durston, a "completed infinity" is a set, but that doesn't prevent us from discussing, treating, or thinking about its members, and there are infinitely many of them.

Next, he says "it is impossible to count to a completed infinity". That is true, but not for the reason that Durston thinks. It is because the phrase "to count to a set" is not defined. We never speak about "counting to a set" in mathematics. We might speak about enumerating the elements of a set, but then the claim that if we begin at a specific time and enumerate the elements of a countably infinite set at, say, once a second, we will never finish, is completely obvious and not of any interest.

Next, Durston claims "one can count towards a potential infinity". But since he defined a "potential infinity" as a procedure, this is clearly meaningless. What could it mean to "count towards a procedure"?

He then goes on to discuss four requirements of an infinite past history. He first asserts that "the number of seconds in the past is a completed countable infinity". Once again, Durston bumps up against his own claims. The number of seconds is not a set, and hence it cannot be a "completed infinity" by Durston's own definition. Here he is confusing the cardinality of a set with the set itself.

Next, he claims that "The number of elapsed seconds in the future is a potential infinity". But earlier he claimed that a potential infinity is a "procedure". Here he is confusing a cardinality with a procedure!

Later, Durston shows that he does not understand the difference between finite and infinite quantities: he claims that "the size of past history is equal to the absolute value of the smallest negative integer value in past history". This would only be true for finite pasts. If the past is infinite, there is no smallest negative integer, so the claim becomes meaningless. So his Argument A is wrong from the start.

At this point I think we can stop. Durston's claims are evidently so confused that one cannot take them seriously. If one wants to understand infinity well, one should read a basic text on infinity and set theory by mathematicians, not agenda-driven religionists with little advanced training in mathematics.

* There are certainly some exceptions to this general rule. The "actual"/"potential" discussion started with Aristotle and hence continues to wield influence, even though mathematicians have had a really good understanding of the infinite since Cantor. Cantor met with resistance from some mathematicians like Poincaré, but today these objections are generally regarded as groundless.

15 comments:

Jeffrey, thank you for taking the time to critique my short article. In reading your critique, I wondered if you had checked the links/references I provided. You will see that some of your key concerns are not with me but with Mathematician Eric Schechter's essay. For example, the word 'procedure' is not mine, but part of his definition of a potential infinity. Also, the definitions and concepts you credit me with are not mine, but Schechter's and I have linked and referenced each one to them. Both you and Schechter are mathematicians, so much of your concern seems to hinge on a disagreement primarily between mathematicians.

"Nearly all research-level mathematicians today (I would guess 99.99% of them) take for granted both "potential" and "completed" infinity, and most probably do not even know the distinction indicated by those two terms."

Wrong, Eric. 100% research-level mathematicians know there is no such garbage in real serious mathematics as "potential infinity" or "complete infinity" and hence no distinction between them. They are both garbage.

I disagree slightly with Dr Shallit: I think wikjpedia is actually an amazingly good source for reading mathematics.

So Kirk says he got his ridiculous claims from a mathematician, not a philosopher...thus he attempts argument from authority. As if that changes the fact that Kirk's writings are full of self-contradictions, contradictions of his own premises, as Jeff has shown. As if a mathematician defines a potential infinity as a "procedure", that makes it OK when Durston cites, then CONTRADICTS, the "procedure" definition. As if it's OK now that Kirk switched the cardinality of a set and the elements in it. As if it changes the fact that the distinction between "potential infinity" and "completed infinity" is still a distinction without a difference, and was still concocted by philosophers, a distinction never derived mathematically from any set of axioms, nor as an axiom ever shown to be useful to derive accurate results. In other words, it's still not a mathematical distinction or concept.

At any rate, here's a previous bit of Kirk's misunderstandings of the basic features of set theory, from his blog.

Kirk Durston: "...let us pretend that the past was actually infinite. Now let us take a quantum leap infinitely far back into the past to some actual past time t. Let us call it t minus infinity. Since we are imagining that the real physical world had an infinite past, that time t minus infinity would actually have had to occur infinitely long ago. Now let us imagine doing what history is supposed to have actually done, and count down from t minus infinity to t now … t minus infinity, t minus infinity plus 1, t minus infinity plus 2, t minus infinity plus three, and so on. When would we arrive at t now? The answer is never. No matter how long we counted down from t minus infinity, there would always be an infinite number of seconds still to go before we got to t now."

Jeff, I think you can catch the problem there: in the infinite set he is considering, he assumes it has element(s) that are infinite. As I pointed out at his blog, the set contains no infinite elements, all elements in it are finite; but the cardinality of the set is infinite. Basic error.

All of this BS because William Lane Craig and Durston require an asymmetry between past and present, hoping to prove by thought alone (no observations) that the past is finite, which then "proves" that God created the universe; but the future is infinite, so they can live forever singing an infinite number of religious hymns in Heaven. Having their cake and eating it too.

Yes, you're right. In that paragraph you quoted, Durston makes the mistake of thinking that the "universe with an infinite past" model entails the creation of the universe at some time t = ∞. But this is clearly erroneous. It just means that at every time t there exists a time t-1 at which the universe existed.

If you want to see how an actual mathematician approaches the same question (with tongue firmly in cheek) see R.K. Meyer, God Exists!, Nous, Vol 21, 1987 -- available on JSTOR. Meyer made many great contributions to systems of "relevant" logic, and always wrote with style and very dry wit.

I agree that "count to a completed infinity" is meaningless, but it is trivial to formalize the notion of counting the elements of an infinite set in a finite amount of time.

To make the notion of counting as concrete as possible, imagine the sequence of positive integers in order, as recited by Count von Count of Sesame Street.

If the Count simply says each number 1% faster than its predecessor (which, while impossible for me, may well be within the powers of a muppet vampire) then the total time should converge (note that the time to recite each integer grows roughly logarithmically so it's a not quite a geometric series) to a finite limit.

For any given integer, it is possible to determine the time at which the Count said it, and all of them would be done in time to complete the episode.

I'm unfamiliar with the philosophical distinction between potential and completed infinity but I don't see how the above example would be considered anything other than a completed infinity.

I loathe people who use "mathematics" to "prove" their religious claims. Kirk Durston is not the worst. At least, one can argue, he doesn't know mathematics. But there are others, veritable mathematicians who keep blowing the horns of mathematics as evidence (or "proof") for the existence of (their) god. Example: John Lennox.

Both types are much worse and much more dangerous than your typical religionist. The latter appeals to the church-goers. The former aspires to attract larger sets of people, those who accept as proof as an argument coming from authority. If only they could understand that Kirk Durston, John Lennox, Jerry Falwell and Jim Jones have similar "logic", they wouldn't be that deluded.

What a confused post. Cantor was a religious crank who brought the contradictory mathematics of infinity into set theory FOR RELIGIOUS PURPOSES (read about him before commenting) and confused multiple generations of mathematicians. You are clearly ignorant of the fact that modern mathematics, with its incomprehensible "continuum," "infinities" and real numbers, is a complete mess.

Tell Cantor to shove his "transfinite numbers," tell Turing to shove his "infinite tape" and tell Godel to shove his "for all." It's all meaningless! These people just hate computers and automation of their boring jobs. We compute with symbols just fine and refuse to do this with runtime logic errors! The law of the excluded middle is nonsense: colorless green ideas sleep furiously. What the hell is the profound nature of finding out that a system could not prove its consistency when you assumed you didn't trust it to start with? This modern theoretical "math" is so BOOOORRRRRRING!!!

How real are real numbers? (Chaitin)http://arxiv.org/pdf/math/0411418v3

You mathematicians try to go down one-way streets with set theory and wonder why you lose information. The theory of algorithms is the future of math and not fuddy duddy set theory that does math backwards and puzzles, like a behaviorist, at the asymmetries.

dimethyl.... Have you actually read the papers you are citing? This rant against real numbers (and infinities) is unjustified. No concrete, simple, alternative has ever been proposed. I'm not sure if Cantor was a religious crank. Maybe he was. But do what? So was Newton. From your phrase "you mathematicians try to go down one-way streets..." I infer that (a) you are not a mathematician and so you may not be able to understand those finite and transfinite numbers and their significance/users, and (b) you have a false impression about what mathematicians do. They don't *try* to go down one-way streets. Sometimes they do, but they don't do it on purpose. Most of the time, however, they go down streets full of interesting questions and problems and, to be sure, aiming at understanding what's out there.